Sr Examen
Integral de 1/(1-x^2/2) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------ dx | 2 | x | 1 - -- | 2 | / 0
$$\int\limits_{0}^{1} \frac{1}{- \frac{x^{2}}{2} + 1}\, dx$$
Integral(1/(1 - x^2/2), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-1/2, c=1, context=1/(-x**2/2 + 1), symbol=x), False), (ArccothRule(a=1, b=-1/2, c=1, context=1/(-x**2/2 + 1), symbol=x), x**2 > 2), (ArctanhRule(a=1, b=-1/2, c=1, context=1/(-x**2/2 + 1), symbol=x), x**2 < 2)], context=1/(-x**2/2 + 1), symbol=x)
Respuesta:
Respuesta (Indefinida)
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// / ___\ \ / || ___ |x*\/ 2 | 2 | | ||\/ 2 *acoth|-------| for x > 2| | 1 || \ 2 / | | ------ dx = C + |< | | 2 || / ___\ | | x || ___ |x*\/ 2 | 2 | | 1 - -- ||\/ 2 *atanh|-------| for x < 2| | 2 \\ \ 2 / / | /
$$\int \frac{1}{- \frac{x^{2}}{2} + 1}\, dx = C + \begin{cases} \sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{2} \right)} & \text{for}\: x^{2} > 2 \\\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{2} \right)} & \text{for}\: x^{2} < 2 \end{cases}$$
Gráfica
Respuesta
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___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\
\/ 2 *\pi*I + log\\/ 2 // \/ 2 *log\1 + \/ 2 / \/ 2 *\pi*I + log\-1 + \/ 2 // \/ 2 *log\\/ 2 /
------------------------- + -------------------- - ------------------------------ - ----------------
2 2 2 2
$$- \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{2} + \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{2} + \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{2}$$
=
=
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\
\/ 2 *\pi*I + log\\/ 2 // \/ 2 *log\1 + \/ 2 / \/ 2 *\pi*I + log\-1 + \/ 2 // \/ 2 *log\\/ 2 /
------------------------- + -------------------- - ------------------------------ - ----------------
2 2 2 2
$$- \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{2} + \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{2} + \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{2}$$
sqrt(2)*(pi*i + log(sqrt(2)))/2 + sqrt(2)*log(1 + sqrt(2))/2 - sqrt(2)*(pi*i + log(-1 + sqrt(2)))/2 - sqrt(2)*log(sqrt(2))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.